When you hover the cursor over an astral, it should show you two numbers where it lists EXP. This should include both a number representing how much EXP the astral has, and how much EXP it requires to level up. See for instance this:
The astral has 0 EXP, but it says that this is out of the 50 EXP that it will need to level up. Hence, it currently has 0 out of 50 EXP needed for the next level. It gains EXP when you synthesise other astrals and energies into it. Hence, if I synthesise one green astral into it, it gains 10 EXP:
However, its stats have not changed yet. To change the astral’s stats, we would need to increase the astral’s level. The astral’s level will increase when it reaches 50 EXP, so if we synthesise 4 more green astrals (10 EXP) into it the astral should go to level 2:
As you can see, the DEF stat granted by the astral has increased. We would now need around 600 EXP to get this astral to level 3. When it reaches the next level, then its stat will increase. However, the astral’s stats won’t change through increasing its EXP until the level increase happens.
The astral basically just increases the stats that it lists, such as ATK in your case. This stat increase gets larger each time the astral levels up. Some astrals increase percentage-based or unconventional stats, such as crit bonus or block rate. A few of those astrals are rare and only available by luck, but the ones for DMG rate and Reduce DMG rate are available in the points exchange. Obtaining astrals gives you points, and these allow you to exchange for astrals of different kinds.