why do teams receive picks if another team signs their free agent?

Its compensation for losing a quality player.

Teams can earn Compensatory picks in the draft based on departing free agents. Free Agents are ranked by the Elias Sports Bureau based on their previous two years of playing, and against players of similar positions. The top 20% of free agents are considered 'Type A' free agents, and the following 20% are 'Type B' free agents. Type A free agents will get their former team a supplemental round pick between the first and second rounds, and a compensatory pick from the signing team. Type B free agents get their former team just a supplemental round pick. To earn a compensatory pick, a free agent must either be signed before the arbitration deadline in early December, or be offered arbitration by their former team but still sign with someone else.

Compensatory picks that one team gives another via this method are the highest available pick that team has. If a team owes two other teams draft picks via Type A free agents, the team whose departing player had a higher score gets the higher ranked pick. A team cannot lose picks it has earned via compensation. Also, the first 15 picks in the draft cannot be lost via compensation, so a team that is in that rank would give up their second round pick.

They negotiate. There are different levels of free agency. Restricted or unrestricted.

Unrestricted free agents can negotiate with any other team without limitation.

A few players, designated unrestricted-first refusal, who would otherwise be unrestricted free agents, have negotiated rights of first refusal with their old clubs on an individual basis. In such an instance the player's old club has a contractual right to match any offer from a new team (within a specific period of time, generally a few days) and thus retain the player under contract.

Restricted free agents (RFAs) can negotiate with any other team, and sign an offer sheet from another team, but their current teams retain a right to match any other team's offer within 7 days of receipt of the offer sheet, in which case they must remain with their current team. If the current team refuses to match an offer, the league can determine appropriate compensation from the new team in the form of draft picks.