This page will give you an understanding of the uses and some strategies of asteroid fields.
For further information on asteroid fields, please check the Dark Forest Wiki.
The main function of asteroid fields is their silver production - they are the only celestial bodies within the game that generate silver.
Asteroid fields have low defense, making them great targets during early expansion and when attacking a players empire - allowing you to diminish your enemies energy quickly, and potentially even stealing any silver which is upon it.
Asteroid fields of higher levels produce silver more quickly than lower levels, and have a higher capacity of silver in total.
You may notice that asteroid fields always spawn with 50% of their silver capacity already filled (indicated by the gold numbers above the asteroid field). This means that if you are the first to take over an asteroid field, that silver becomes yours, and you can then use it to either upgrade a planet you own, or to withdraw for score (if that is a part of the rounds scoring mechanism).
Asteroid fields have a multiplier of 1x for most of the default stats, except for defense and silver capacity. Their defense is only 0.5x of the default value and their silver capacity is 2x the default value.
Due to their reduced defense, asteroid fields can serve as a great way to quickly get your empire expanding, as less energy is required to initially seize the asteroid field from pirates. It can also be useful to get a few asteroid fields early to begin generating silver in preparation for several quick upgrades in succession once you do start taking over planets. After you begin taking over planets of higher levels however, low level asteroid fields quickly lose their effectiveness, as the silver generated is often to slow for large planets - they can then be ignored or abandoned after you feel they are no longer useful.