means The process of becoming warmer, or of someone gradually growing fonder of or more receptive to something.

from From Old English 'wearm' (warm), tied to a deep Germanic root and possibly a cousin of words for fire and burning across the Indo-European family. The '-ing' simply turns the state into an action — the becoming-warm. The figurative sense of 'warming up to' someone or some idea — that slow thaw of feeling — is a later, homely extension, as natural as creeping toward a hearth.