The costs will probably reach several thousand dollars.

Literal

Costs [topic-は] several-thousand-dollars [even to-にも] reach [probably-だろう].

何千ドルにも達する means 'to reach as much as several thousand dollars.' 何千 means 'how many thousands' = 'several thousand.' ~にも adds emphasis: 'even to that amount.' 達する (たっする) is a formal verb for 'to reach, attain.'