If my uncle passed away and he told me he was leaving me something, how do I find out about his estate?

If there is a Will, "interested" parties may ask to view copies of it. An interested party is anyone who is an immediate family member of the deceased, whether or not they are listed in the Will. The same applies to anyone who is listed in the Will as a beneficiary. Depending on the circumstances, other parties may be able to demonstrate they have enough of an interest in the Will to request a copy of it. 
The easiest way to obtain access is to ask one of the people who either has a copy or who can get one (i.e. the attorney who drafted it, etc.) since they are obligated to give it to you if you’re legally entitled to it. You can also make a formal request to the applicable probate court  to view the Will. Additionally, once it has gone through probate, the information within it becomes public record and anyone who chooses to search for it can see it. 
If there is no Will, your uncle died "intestate". This means that the intestacy or "succession" laws of the state in which he was domiciled as of the date of his death will control. Typically, such an estate is divided between the survivng spouse, if any, and the children of the deceased. If your uncle had no spouse or children then his siblings, if any, would be next in line, then his parents and then nephews and neices. You can check on the estate with the probate court for further information.