How does the LHC explore extra dimensions?

First, no evidence for other dimensions has been found.

However, there are ways for particle colliders to detect other dimensions. One of the main ones is to see if any energy "disappears" under very certain circumstances...then it could've possibly gone into another dimension.

Another way is to look for particles that can only exist if there are other dimensions. These particles would be around 100 times the mass of the W and Z bosons (these carry the electroweak force). Particles of this size could really only be detected by the LHC.

Other ways include examining the evaporation of mini black holes and seeing what particles are produced and looking for gravitons - gravitons would quickly disappear into extra dimensions, leaving an apparent loss of energy, which could be detected.

I'll be updating this as I find more information, though these are some of the main ways. Hope this helps!

This website has more information.