How can I fix my lower implant to stay snapped in place? (Photo)

Changing the plastic attachment within the denture could help, but often the abutment itself may be worn and may need to be replaced as well.
The real problem though is quite likely to be the occlusion (the way the bite comes together). It will need to be adjusted in a way that when you chew and move your jaw left to right, the dentures by way of having a balanced occlusion, can be more retentive.
If occlusion is the real problem, then there will always be accelerated wear of the components, and even if either the plastic insert or the abutment itself get changed, the issue is likely to recur.