Hearts, like all common pickups, can be gained by checking some places (barrels, plants, etc.), dropped by dead enemies, or bought in shops. Another source of life points is the Horn of the Blue Triton, which is capable of storing and dispensing them.
In both LBA and LBA2, Twinsen has 50 hearts' worth of maximum health for the whole game. The Horn of the Blue Triton can store up to 100 hearts, dispensing them in 5-units.