Learn Russian: Почему ты режешь мясо таким старым ножом? - Why are you cutting the meat with such an old knife?

They are in the Instrumental case because the knife is the tool being used to cut the meat. In Russian, you don't need a preposition like 'with' when using a tool; you just put the noun and any adjectives describing it into the Instrumental case.

Why does таким end in -им but старым ends in -ым? They both describe the masculine noun ножом.

Both use masculine Instrumental endings, but a spelling rule forces them to look different. The standard 'hard' ending is -ым, which we see in старым. However, the stem of такой ends in к. The spelling rules say you can never write the letter ы after к, г, х (or the hushing sounds). Therefore, the ending has to be spelled with an и, giving us таким.

Why does the word for knife (нож) become ножом here?

Standard masculine nouns taking the Instrumental case add -ом. There is a tricky rule for words ending in 'hushing' sounds like ж, ш, ч, щ, and ц: they only take -ом if the stress falls on the ending itself, as it does in ножо́м. If the stress falls on the stem, the ending becomes -ем (for example, му́жем — with a husband).