In one word, what do the structs in ge.h do?

Not in one word, but the structs defined in ge.h are different representations of a group element (aka a point on an elliptic curve):

/*
ge means group element.
Here the group is the set of pairs (x,y) of field elements (see fe.h)
satisfying -x^2 + y^2 = 1 + d x^2y^2
where d = -121665/121666.
Representations:
  ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
  ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
  ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
  ge_precomp (Duif): (y+x,y-x,2dxy)
*/

source.

Monero uses keccak...

Which has nothing to do with elliptic curves (or by extension groups); keccak is a hash algorithm.

In one word, what do the structs in ge.h do?

Not in one word, but the structs defined in ge.h are used to store different representations of a group element (aka a point on an elliptic curve):

/*
ge means group element.
Here the group is the set of pairs (x,y) of field elements (see fe.h)
satisfying -x^2 + y^2 = 1 + d x^2y^2
where d = -121665/121666.
Representations:
  ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
  ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
  ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
  ge_precomp (Duif): (y+x,y-x,2dxy)
*/

source.

Monero uses keccak...

Which has nothing to do with elliptic curves (or by extension groups); keccak is a hash algorithm.