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In tournament Magic, once time is called in a round, the game stops and the players have a total of 5 "extra turns". If, after those five turns, the game is still not over, the match ends in a draw.

Why are there five extra turns? This rule seems like it gives the player who takes the first extra turn a large advantage - after all, that player has three more turns, while the opponent only has two. Why not six or some even number?

The imbalance is intentional. It gives every player an incentive to have the time limit hit during their opponent's turn, and therefore to finish their own turn quickly. As a result, it helps each tournament round stay within the time limit and reduces the need for turn extensions.

From the Tournament rules (pdf)

2.5 End-of-Match Procedure

If the match time limit is reached before a winner is determined, the player whose turn it is finishes their turn and five additional turns are played in total. This usually means that one player takes three turns and the other two, but a player taking extra turns may affect this. If a player has already passed priority in their end step when the time limit is reached, that is considered to be in their opponent’s next turn.

When the time limit hits, the currently active player finishes their turn and, as you correctly note, is disadvantaged by getting only two extra turns vs. the opponent's three. That means you do NOT want to be the active player when the time limit hits. The best way to not be the active player is to finish your turn, so you have an incentive to finish your turn as quickly as possible and pass that "hot potato" to your opponent.

Additionally, with an odd number turn extension, the non-active player also gets the last turn. In a damage race situation with creatures on both sides, the player who has the last turn does not have to plan for the opponent's backswing, and can go all out with the last attack.

By providing each player an individual incentive to play quickly (in addition to regular slow play infraction penalties), the whole tournament benefits. If the turn extension was an even number, no player would have this particular incentive to play quickly, because it wouldn't matter who the active player is at the time limit. The active player would also get the last turn, so it would be an overall benefit to the currently active player and therefore an incentive to play slowly.

The thing is, both players get 3 turns "after" time is called. The active player, or "Turn zero", as it were, will take the current turn, plus turns 2 and 4. The other player takes turns 1, 3, and 5. Yes, it is possible the current turn is only a partial turn if time is called, for example, during combat (or in the worst case, during the end step). It is also possible time is called during the untap step at which point the player is indeed getting a full turn. To account for the latter possibility, the number of extra turns is odd. In an alternate universe, the rules could account for the former possibility, in which case the extra turn count would be even.

As for the secondary question (asked in the comments), and disclaimer this is purely my opinion, it's likely because the clock is not always easily visible to all players, and "Ok, players, everyone synchronize their watches, ready, one, two, three, GO!" is not really practical in a tournament setting of any scale. The idea is that players who don't have access to the round clock have a bit of lead time to finish up whatever they're doing and aren't severely punished if they are, say, mid-sentence moving to lethal combat when time is called (to take an extreme position, while also recognizing that less extreme cases exist).