They still want to get into the low cost segment but want to market themselves differently for better penetration.
You can only present so many images of a brand to customers: You can't show both the super-cheap prices with happy youth on them and posh first class at the same time.
They do not want to tarnish a nice brand by mixing it with a low cost carrier and what that is perceived to entail.
For instance, Germanwings got a somewhat infamous reputation recently, and Lufthansa is probably happy their name was not more frequently used in media.
Low cost is also associated with less good customer service and hidden fees.
Crew are often less well paid, and they are sometimes put under a separate working agreement and company to try to prevent conflicts.
Some legacy carriers have very good pay which is difficult to reduce.
Management does not want to be bound by these when expanding or adjusting their business.
Each case is pretty specific depending on exactly they are trying to do and achieve.
For many American Carriers and Lufthansa, the last one is probably the principle reason.
aviation.stackexchange.com/users/790/Thunderstrike
As mentioned there are lots of branding issues around this
but there are also lots of corporate and legal reasons to do so.
Business Isolation : If you want to start out a bunch of new routes and you are not sure how well they are going to do a large airline
my not want to book that kind of loss or risk having to cover it.
Thus they can incorporate a "new" airline and protect their existing and profitable business interests should the "new" airline go under.
Likewise it also makes it easier, should it be very profitable, to potentially sell it off (or absorb it as well).
Taxes/Fees :
Tax law changes from place to place so this may differ depending on where they are based but if there are different taxes as a result of company size or specific aviation taxes they may be able to take advantage of them by splitting up.
Legal Isolation : I am no lawyer
but I am sure that how ever they set these up
it isolates the parent airline should anything happen to the other carrier.
Image/
Branding
: this has been touched on already but as mentioned you may not want the image that comes with running a low cost brand associated with your luxury brand.
aviation.stackexchange.com/users/3677/Dave
I would imagine this has to do with branding.
Kind of like how some car companies have both "regular" and "premium" brands.
Eg., Chevy v. Cadillac or VW v. Audi, etc etc.
It allows for companies to have a more exclusive version of their product for people who are just the premium, exclusive sorts...
Granted
there are other cost considerations, and other branding considerations, and I could probably sit here and go on for hours with that
but I think it would be a little bit outside of scope.
So, to just give a basic explanation:
Different brands have different associations with them.
Some are seen as premium, exclusive, indulgent.
Others are seen as just hitting the basics.
And some companies will avoid mixing the two just for the sake of brand identity in marketing.
aviation.stackexchange.com/users/511/Jay Carr
As mentioned in earlier answers, branding and management (pay, collective bargaining agreements, etc.) are certainly two major factors.
And legacy airlines are mindful of the risk to see their profit cannibalised by the new cheap fares so they need to maintain as much segmentation/price discrimination as possible.
Interestingly, the two strategies are not exclusive.
Air France/KLM has several low-cost subsidiaries (Transavia, Transavia-France and HOP!)
but also introduced many low-costs tactics (no meal, surcharge for hold luggage…) on short-haul European routes on both legacy airlines.
The thing is that some elements of the low-cost model (paid refreshments, quick turnaround, routes to cheap airports or use low-cost terminals at major airports, single-class cabin with as many seats as possible, identical aircrafts…) can only be introduced for the whole airplane or even the whole fleet.
And for a highly price-sensitive market segment, merely using some elements of the model to offer lower prices is not enough, you need to go all in to ensure your fares are as low as those of your direct competitors (say Easyjet, not necessarily Ryanair).
aviation.stackexchange.com/users/3018/Relaxed
A business reason no-one has mentioned yet is the original reason the big names started: Because low cost airlines began flying routes cheaply and created a market If the big names hadn't joined in, all that revenue would have gone to the low cost airlines, who would have grown to the point they could potentially compete in the existing markets, so it was essential that the big players joined in.
It is not possible to run a low cost airline in the structures of a major airline (the cost model is entirely different - passenger demographics, services on board, pre-flight hospitality, additional benefits etc.)
so the only way to do it that makes financial sense is to run it as a separate logical entity.
It has proven to be a difficult change for some of those big airlines, with various attempts at creating budget versions failing, but some have managed to compete successfully.
aviation.stackexchange.com/users/59/Rory Alsop
There are various full service carriers (FSCs) who have a separate low-cost carrier (LCC) subsidiary. For example:   Germanwings  by  Lufthansa  Air India Express  by  Air India  JetKonnect  by  Jet Airways   Why do the FSCs prefer to launch a separate LCC, rather than providing seats at a cheaper rate with reduced comforts within their existing structure?
Why do full service airlines prefer to run a separate low-cost carrier?
aviation.stackexchange.com
<commercial-aviation><airline-operations><low-cost-carrier>
aviation.stackexchange.com/questions/17340
aviation.stackexchange.com/users/8482/anshabhi
Brand and imaging are major factors in large airlines running a separate low-cost airline.  The two types of airline offer very different services and appeal to a different target group. If large airlines did not run a low-cost airline, they would have lost all those customers to other players in the market. Low-cost airlines are typically structured completely differently from major airlines and thus it would be impossible or undesirable to run the two sides as one company. The main reason seems to be branding. It makes more sense to separate an exclusive brand from a low-cost brand

Management is also a factor. The structures of large airlines do not lend themselves to low-cost models. Separating the companies may be beneficial in terms of salary and taxes. From an industry perspective, it made sense for the larger airlines to become involved in the low-cost segment so that low-cost airlines didn't dominate the market. 
It comes down to branding. It makes more sense to separate an exclusive brand from a low-cost brand rather than attempt to combine the two. Low-cost airlines are associated with lower-quality customer service and hidden costs. Management is a factor - low-cost airline crew are often less well-paid, and separate companies may help to prevent conflict. From a tax perspective, it may be beneficial to have two separate companies. From an industry perspective, it made sense for the larger airlines to become involved in the low-cost segment. If they hadn't, the low-cost airlines may have been able to dominate the market. The structures of large airlines do not lend themselves to low-cost models.