Learn Arabic (Modern Standard): أحسست بألم في قدمي.

This is exactly how doubled verbs work in the past tense! The "he" form of the verb is أحسَّ (aHassa), where the shadda shows a doubled س. But when we add the "I" suffix ـتُ (-tu) to say "I felt", that doubled letter splits into two separate letters: أحسَسْتُ (aHsastu).

The root for this verb is ح-س-س. Why does أحسست start with an أ (alif with a hamza)?

This verb belongs to a slightly more advanced verb pattern (Form IV). The base verb is أحسَّ (to feel or sense). Past tense verbs in this pattern always start with an أ (alif with a visible hamza on top), which remains part of the word when we conjugate it, such as in أحسستُ.

Why is the preposition بـ attached to the word ألم (pain)?

The verb أحسّ (to feel) is almost always followed by the preposition بـ (bi). In English, you say "I felt a pain," but in Arabic, you say "I felt with a pain" (أحسست بـألم). Whenever you use this verb to describe feeling something, remember to include that required بـ!

The word قدم (foot) doesn't have a taa marbuuTa (ة). Is it masculine or feminine?

It is feminine! In Arabic, body parts that come in pairs (like feet, hands, eyes, and ears) are almost always grammatically feminine, even without a feminine ending. Here, we just added the possessive suffix ـي (my) to make قدمي (my foot).