Learn Arabic (Modern Standard): أنا سافرت إلى مصر الشهر الماضي.

In Arabic, past tense verbs change their ending to match the person doing the action. The suffix ـتُ (-tu) at the end of سافرت is what tells us it was 'I' who traveled. The base verb without the suffix is سافر (he traveled).

Since the verb already means 'I traveled', why do we also have the word أنا (I) at the beginning?

You actually don't need it! Because the suffix on سافرت already means 'I', saying سافرت إلى مصر is perfectly complete and common. Adding أنا here just gives a little extra emphasis to the fact that you were the one who traveled.

Why are there two words with 'the' (الـ) to say 'last month' (الشهر الماضي)?

In Arabic, 'last' (الماضي) acts as an adjective describing the month. A major rule you learned earlier is that when a noun has 'the' (الـ), any adjective describing it must also have 'the' to match. So literally, this translates to 'the month the past'.

Why doesn't مصر (Egypt) have 'the' (الـ) before it?

Many Arabic country names don't use 'the' because they are already specific names. While some countries like Yemen (اليمن) include it as part of their official name, مصر never takes الـ.