Scenario 1:
It is round 2 of a 6-player game, so 3 players are needed for the mission.
Alice (loyal) is the leader and strongly (but silently)
suspects Bob of being a traitor, perhaps due to a "tell".
Carol also suspects Bob and is strongly saying so, and Bob is playing offended.
Alice nominates all three of herself, Bob, and Carol for the mission.
She knows that Carol will vote no and will vote no herself, so if even one other member is skeptical the vote will fail.
Meanwhile, Alice carefully watches the votes from Donald, Elaine, and Frank.
A yes vote, especially a single yes vote, is a suggestion that one of those players is a co-spy; not an ironclad proof, but a possibly-important hint.
Scenario 2:
Same situation, but in this game Alice is a spy.
Carol's suspicions about Bob are baseless.
Alice still goes through the motions of this charade even though Bob and Carol are both loyal, to throw off suspicion that would arise because she is not going through her typical plan.
boardgames.stackexchange.com/users/3278/sitnaltax
Sometimes it works as an information-gathering gambit.
You nominate a team, then vote it down hoping it fails, but taking note of who voted for it.
What to make of those votes of course depends a lot on playstyle and whether the vote passed or not.
If you are under heavy suspicion, you might either pull the above gambit or pretend to just so you don't take anyone down (or in the case of a spy, take the wrong people down) with you.
To manipulate the turn order.
If using The Plot Thickens, you might intentionally tank a vote just to manipulate who gets plot cards on a future turn.
In particular, it lets you skip over a suspected spy if the team can still afford to vote down two teams in a row.
Speaking of which, this can also be a prelude to a spy victory by five failed votes (No Confidence and Strong Leader can set this up)
boardgames.stackexchange.com/users/6587/Free Monica Cellio
At the start of the game, the resistance has the advantage of numbers and the spies have the advantage of knowledge.
The core tactic of the resistance is to reduce the spies' advantage by making as much knowledge public as is possible.
One of the biggest pieces of public knowledge is what teams are proposed, and how people vote for them.
As such, in the groups that I play with, it's extremely common in the first round for leaders to genuinely propose teams, and then vote against them.
In addition, if the game comes down to the last mission, then obviously resistance needs a perfect team to win.
Hence any resistance member not on the team must vote against it, as should any resistance member who is on the team but suspects that there are still spies on it.
If there are two resistance members coming up as consecutive leaders (and they're reasonably convinced of each other's alignment), then the first can propose a team that includes both of them on it, and vote against it.
If anyone not on the team votes for it, that's pretty good evidence they've just outed themselves as a spy, otherwise you can feel more confident about the team and the next leader can propose the exact same team.
boardgames.stackexchange.com/users/5356/ConMan
I thought I'd add a scenario I thought of.
Alice is the leader.
Bob is accusing Charlie of being a spy.
Alice suspects that Charlie isn't a spy, but Bob is, and is trying to sow seeds of discord.
Alice, without making a deal of it (pretends to ignore/
not notice Bob's arguements)
assigns Alice, Bob and Charlie to the team.
If Bob really was Resistance, and was genuine about accusing Charlie, then Bob will vote against this assignment.
If he votes for it, he probably is a spy.
boardgames.stackexchange.com/users/7687/dwjohnston
To counterpoint the other answers, which are all great.
The leader might be the sort of person that has a policy of voting down every 1st (and 2nd, and 3rd) proposal on a mission.
The leader might be using some random or algorithmic method to determine how to vote.
Yes - I've seen these and more...
boardgames.stackexchange.com/users/2040/xorsyst
In a  previous question  a commenter mentioned that team leader might propose a team, that they don't intend to approve.  Presumably this has something to do with either seeing how other people vote to ascertain if they are spys or not, or sowing discord if the leader is a spy themself.  Can you provide some specific examples where a leader might do this?
Why might a leader propose a team they don't plan to approve?
boardgames.stackexchange.com
<the-resistance>
boardgames.stackexchange.com/questions/16723
boardgames.stackexchange.com/users/7687/dwjohnston
Sometimes this is used to gather information or the leader might be using a random voting method. This is sometimes used an information-gathering gambit. The leader might be the sort of person who likes to vote down every proposal on a mission ormight be using a random or algorithmic method to determine how to vote.  
This is sometimes used an information-gathering gambit. The leader might be the sort of person who likes to vote down every proposal on a mission ormight be using a random or algorithmic method to determine how to vote.  