Every time I see a photo of my late mother, something warm wells up in my chest.
Literal
Late mother [possessive-の] photo [object-を] see [every-time-たびに], chest [dat-に] hot thing [subject-が] well-up-come
Two rich pieces of idiom. ~たびに attached to a verb's dictionary form means 'every time X happens'. 胸に熱いものが込み上げてくる is a stock emotional phrase — literally 'something hot surges up into the chest' — used for being choked up by grief, joy, or nostalgia. 込み上げてくる uses ~てくる to mark the feeling coming toward the speaker in time, intensifying the sense of emotion overtaking them.