Question

How does the timer ball work?

Answer 1

Gens 3 to 4: modifier = min((turns + 10) / 10, 4)
It effectively raises the modifier by 0.1 every turn until it reaches 4 at turn 30.
Gen 5 onward: modifier = min(1 + turns * 1229 / 4096, 4)
It effectively raises the modifier by roughly 1/4 each turn until it reaches 4 at turn 10.
source

Answer 2

The Timer Ball (Japanese: タイマーボール Timer Ball) is a type of Poké Ball introduced in Generation III. It can be used to catch a wild Pokémon, being more likely to succeed the longer it has been since the start of the battle.

So basically, the more further you are in a battle, the better the Timer Ball works.

The formula for catching a Pokémon in this Ball is here.

Please correct me if I'm wrong.

Answer 3

The Timer Ball has a higher success rate the more turns have been taken in battle. For example, it would be more likely to successfully capture the Pokémon if your Pokémon had used a move multiple times. Here: https://bulbapedia.bulbagarden.net/wiki/Timer_Ball#Manual_activation