I got gum stuck to the bottom of my shoe.

Literal

Shoe [possessive-の] underside [on-に] gum [subject-が] stubbornly-stuck-[regret-ちゃった].

こびりつく means 'to stick stubbornly, to be caked on' — stronger than くっつく, implying the gum is really stuck and hard to remove. ちゃった is the casual contraction of てしまった, conveying mild dismay at an unfortunate result. A perfectly natural everyday complaint.