There are various reasons: Societal mechanisms - Lower levels of trust.
This is a common social approach -
if someone shows themselves as willing to break the social contract (laws), they have lower level of trust.
In terms of franchise, it means they can't be trusted with the responsibilities of said franchise - as a concept, not necessarily out of specific "he will vote for a felon organization" fear.
Moral - Punitive.
Leaving aside whether this is a good idea, or an efficient one; there are definitely people who feel that removing franchise for committing a felony is a just punishment.
Practical - worry about abuse of power.
This is slightly similar to #1, but whereas #1 was a generic principle over generic trust levels, this one is more about specific consequences.
To counter your somewhat amused "felons organising politically and exerting an influence on politics" quip, recall that this is exactly what organized crime and corruption does and achieves, from Sicily to Russia.
Or going further back, Blues and Greens of Rome.
politics.stackexchange.com/users/115/user4012
From this answer on Why allow convicted criminals to vote? , @origimbo gave one reason to allow them to vote as being protection against the following algorithm: Win a legislative election.
Pass any law which disproportionately imprisons the supporters of your opponents.
Profit.
conversely, one reason to keep them from voting would be to use this algorithm. @Schwern
posted in the comments of the original answer pointing out that the War on Drugs was one case where this was used.
politics.stackexchange.com/users/21078/Alexandre Aubrey
You mentioned this practice in relation to the US; the legal basis at least would be section 2 of the 14th amendment :
Representatives shall be apportioned among the several states according to their respective numbers, counting the whole number of persons in each state, excluding Indians not taxed.
But when the right to vote at any election for the choice of electors for President and Vice President of the United States, Representatives in Congress, the executive and judicial officers of a state, or the members of the legislature thereof, is denied to any of the male inhabitants of such state, being twenty-one years of age, and citizens of the United States, or in any way abridged, except for participation in rebellion, or other crime , the basis of representation therein shall be reduced in the proportion which the number of such male citizens shall bear to the whole number of male citizens twenty-one years of age in such state.
Felony disenfranchisement was specifically upheld because of this in the case Richardson v. Ramirez .
The legal basis of the practice in the US is sort of an anachronism since a major reason the amendment is worded that way was to help maintain Republican power and suppress the Democrats who were previously part of the Confederacy.
So the historical case that actively grants the state this power, came about because the Republicans wanted to more easily maintained political power.
This seems like an obvious avenue for them to take since the power to curtail (and expand) voting rights is pretty powerful in a democracy.
It granted a legal basis to Republican rule beyond just military occupation of the South.
It's hard to judge the intent of the practice in all individual cases since laws regarding felony disenfranchisement were passed before and after the 14th amendment and the Reconstruction era.
But given the history of Jim Crow and poll taxes, literacy tests, and grandfather clauses meant to restrict voting rights, this practice seems like another obvious way to control voting rights and maintain white supremacy.
Sometimes its really obvious, like the 1901 Alabama state constitution that removes voting rights for among other things, "miscegenation", which of course, is itself a crime.
Other times we have to look at the results to give us clues about what the intent may have been.
For example, when we examine the relative rates of current felony disenfranchisement and where it occurs, the desire to maintain political power and white supremacy via control of voting rights is probably still present:
politics.stackexchange.com/users/16221/Teleka
So that you can suppress voting among a constituency that does not support you.
Look at the group of people that is disproportionately incarcerated and what are typically their voting intentions .
If you crunch the numbers, you will find that for every single Republican vote removed by felony disenfranchisement, you remove roughly five Democrat votes.
politics.stackexchange.com/users/16523/Oscar Bravo
The scholarship of Montesquieu is particularly apt here :
According to Montesquieu, a necessary condition for the existence of a republican government, whether democratic or aristocratic, is that the people in whom supreme power is lodged possess the quality of “public virtue,” meaning that they are motivated by a desire to achieve the public good.
Although public virtue may not be necessary in a monarchy and is certainly absent in despotic regimes, it must be present to some degree in aristocratic republics and to a large degree in democratic republics.
Sounding a theme that would be loudly echoed in Madison’s
“Federalist 10,” Montesquieu asserts that without strong public virtue, a democratic republic is likely to be destroyed by conflict between various “factions,” each pursuing its own narrow interests at the expense of the broader public good.
From him directly : When virtue is banished, ambition enters the hearts that are disposed to receive it, and avarice enters all of them.
The desires change their objects; what once they loved, they love no longer; they were free under the laws, and now they want to be free to act against them; each citizen is like a slave who has run away from his master’s house;...
The republic is reduced to mere appearance; and its strength is but the power of a few citizens, and the licence (to commit crime) of all.
Allowing criminals to vote diminishes both civic bonds and what it means to be a citizen precisely because it allows the contagion or criminality a voice and entry as legitimate into the public sphere.
politics.stackexchange.com/users/9790/K Dog
Felony disenfranchisement  is to take away someone's right to vote during or after this person is in prison.  It appears particularly common in the United States.  What is the aim of such a rule?  Is it meant as an additional punishment to the individual involved, or is there a perceived risk of felons organising politically and exerting influence on politics?
What is the motivation for felony disenfranchisement?
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<voting><justice><disenfranchisement>
politics.stackexchange.com/questions/7840
politics.stackexchange.com/users/130/gerrit
There are moral and punitive reasons. However, there may be practical reasons, with concerns about abuse of power or profiteering. Some say it may be used to suppress voting among a constituency that does not support you. The reasons are as follows: moral, punitive, to prevent abuse of power, to suppress voting or to profit.
The reasons are as follows: moral, punitive, to prevent abuse of power, to suppress voting or to profit.