Let the diagonals intersect at O.
Rhombus => AB = BC = CD = DA and diagonals AC and BD are perpendicular.
compare the triangles ABO and BOC.
angle AOB = 90 = angle AOC.
side AB = side AC
side OB is common.
hence they are congruent. So angle ABO = angle CBO
hence diagonal BD bisects angle B.
You will find similarly, AOD and DOC are congruent. There is one common side, one right angle and AD = CD. Hence OD (or BD) bisects angle D.
Similarly triangles BOC and COD are congruent. Hence OC (or AC) bisects angle C.
similarly, from triangles DOA and AOB , we find OA (or CA) bisects angle A