u/mignyau gave a pretty good thorough explanation above, but a couple points of clarification:

hieroglyphs - The Japanese script you're likely referring to are called "kanji" and are technically considered "logographs". Hieroglyphs are a very different type of symbol.

Totals - there have been WAY more than 1,000 kanji in existence throughout the Japanese language's many years of standardizations and revisions (there is no definitive number). However, a little over 3,000 kanji are readily used in common communication nowadays.

Typing - most people typing in Japanese today use an input method called "romaji nyuuryoku" (romaji input). "Romaji" is a method of writing Japanese that uses Latin script (a,b,c's) to write out Japanese. For example, "arigatou" is the romaji writing of ありがとう. A piece of software on most Japanese computers/devices then converts the romaji into kana (Japanese phonetic script) or in some cases straight into kanji where appropriate. That's the basics of it, at least.

As with most things that you grow up doing in your life, you get used to it with enough repetition.