When your axies reach a certain amount of AXP, they will get an increase in their level, which is an on-chain attribute that is added to your axies. As AXP gain is off-chain, you will need to perform a transaction to sync your Axie's level on-chain at every 10 levels through a process called Ascension.
For every 10 levels worth of AXP that your axie gains, you must ascend your axie before they'll be able to gain further AXP and levels. The Ascension process involves submitting an on-chain transaction, which syncs your axie's on-chain level attribute to their current level.
There are 6 ascension points at the moment [levels 9, 19, 29, 39, and so on].
1st Ascension: 1 to 10 > 13,310 total AXP needed, costs $0.50 of AXS to ascend
2nd Ascension: 10 to 20 > 110,480 total AXP needed, costs $1.00 of AXS to ascend
3rd Ascension: 20 to 30 > 334,170 total AXP needed, costs $1.50 of AXS to ascend
4th Ascension: 30 to 40 > 701,300 total AXP needed, costs $2.00 of AXS to ascend
5th Ascension: 40 to 50 > 1,223,670 total AXP needed, costs $2.50 of AXS to ascend
6th Ascension: 50 to 60 > 1,910,670 total AXP needed, costs $3.00 of AXS to ascend
Note: At the moment, you can only ascend your axie six times. This means that it will no longer gain any more AXP after it gets to Level 60, and cannot be ascended further until higher levels of ascension are unlocked.