To my knowledge, people horse deities to become closer to said deity, share energy with the deity, or carry out a task for him/her. I'm sure there are several other reasons that people do it, but those are the ones that I see or hear the most.
In addition to becoming closer to a deity, horsing can be done for devotional work (if you have a relationship with a deity, it'll sometimes happen involuntarily). It can also be done for acting as a medium of adeity for a congregation.
It has other names as well; mediumship (a form of it), invocation, possession (voluntary and involuntary), etc.