a small token; a little something
Humble expression used when offering a gift or doing a favor, downplaying its value. Often used in set phrases like 気持ち程度ですが or 気持ち程度のものですが.
It's just a small token, but please accept it.
It's nothing much, but please take it.
Compound of 気持ち (feeling) and 程度 (degree, extent). Literally 'to the extent of a feeling', used to humbly minimize one's own offering.