It begins with the statement that Pinchas ben Elazar ben Aharon the priest turned away the Divine anger from the people of Israel (Num. 25:10). And anger is not the only emotion which the Bible attributes to the Him. There are positive emotions like love and joy, and negative emotions like jealousy and hatred.
That they combine to give us a picture of a God who feels, is perfectly obvious. Yet they create a series of theological problems. One is the implication that God is not constant or consistent, that He can move from one passion to another.
A more serious problem is whether emotions are possible at all with a God who has no shape or form, who is not and cannot be affected by physical, psychological or other events like His human creatures can and do.
Something hurts me as a human: I react with pain or even stoicism. Something gives me pleasure: I respond with a smile. This is the nature of earthly, human life. It makes perfect sense. This is the way we are made. However, God is above and beyond such events.
Even if you remove the physical metaphors from the Biblical references such as His nostrils quivering and His heart rejoicing, you still have a problem.
The approach of the sages was to say, Dibb’rah Torah kil’shon b’nei adam, “The Torah speaks in the language of human beings” (B’rachot 31b etc.). When we want to use language to speak of God, our only language is human. Rationally we know that our language is too limited to apply to God who cannot be confined within sentences or defined within linguistic boundaries. All we can do is as the rabbis did, to say, kiv’yachol – “as it were”.
God does not have human emotions, but what else can we do when we speak about Him? We are speaking in metaphors, in poetry, not in factual statements. If we had to restrict ourselves to factual statements we would be unable to say anything.