I am trying to use the XY Table to Point operator to create point features; however, it doesn't appear on the map. My end goal is to intersect these point features with a polygon layer, but the projection or spatial location of the points doesn't show up. This causes the intersect between the points and the polygons to return an empty set. What is the proper way to convert LAS points to point features?

LAS To Multipoint—Help | ArcGIS Desktop

Shows as the option, then you could use Multipart to Singlepart tool

If the coordinates are just in a table, you would have to provide the coordinate syste.

What is the projection of the point file (assuming it became points)?

If the coordinate system is undefined, you need to define it using the Define Projection tool.

If you then need make it some other coordinate system, then you use the Project tool in arctoolbox to get files into a common coordinate system