Why do we drink water while eating food? We do this so that the solid, nutritious food taken in by us gets digested and absorbed properly and in turn we derive good health, peace and comfort.
Similarly, why do we perform Puja? We worship God for our own contentment. The intention behind this is to acquire good mental health devoid of impurities and thereby to lead a broad-minded, contented and peaceful life.
Some people casually listen to the radio to get mental peace. There are others who, for the same reason, simply walk up and down aimlessly. There are also people who, not knowing what to do, simply pick up a quarrel with their neighbours or family members. There are yet a few who use their idle time to recollect all the past events of their life thus causing great harm, both to themselves as well as to their mind. They sit gloomily. This is their fate. Recollecting events that happened decades ago and by constantly thinking about it, they are only harming themselves. With this routine, eventually the body as well as the mind gets diseased.
Your question was about the purpose behind the performance of pujas, abhishekas and homas. The various mental illnesses that arise out of an idle mind, like the one mentioned above, that cannot be subdued through any form of medication and treatment can be healed completely through worship. Due to worship the person enjoys mental peace.
-Glimpses of Bhaktimala 4