EMB Legacy 500 pilot reported on arrival having a miscommunication with ATC resulting with the pilot descending below their assigned altitude and flying towards terrain. ATC issued a low altitude alert and new clearance.

Narrative

While descending on the ZZZZZ arrival northwest of ZZZ we requested and received vectors for the ILS XX at ZZZ. On a northern heading ; the Approach Controller requested best rate of descent; directed a right turn to 080 heading and descent to 5000' msl (I thought) and to slow and maintain 170 knots. The PIC (pilot not flying) was on another radio talking to the FBO; so I repeated instructions back to Approach Control as 'right turn heading 080; descend to 5000 and 170 knots.' I did not hear a correction to the 5000 feet msl altitude after my read back. While I was descending through approximately 5400 feet msl and slowing in VMC with airfield in sight; Approach Control issued a 'low altitude alert' and said my clearance was to 6000 feet msl and asked if I had field in sight. I replied yes and leveled off. Approach Control cleared us for the visual approach to Runway XX at ZZZ and switched us over to Tower frequency. We executed the visual approach to Runway XX to a safe landing and taxied to parking at which time Ground Control gave us a number to call in response to a possible altitude violation.The radios were understandably busy in this terminal area and I believe I heard 5000 feet as my altitude clearance but 6000 feet clearly makes more sense. I do not think I read back 6000 feet and don't believe I heard a correction if I did read back 5000 feet msl. I was VMC with field in sight; no other aircraft were affected by this sequence; and I safely executed the visual approach to Runway XX.