Lanthanum chloride reacts with the free, dissolved phosphate compounds to form a salt (precipitate) of Lanthanum and phosphate (lanthanum phosphate) which is not soluble in water. The chlorine ion from the Lanthanum chloride likely ends up bound to free sodium ions to form sodium chloride, table salt, which would dissolve in water. Lanthanum sulfate is also in these phosphate removers and reacts in a similar way with phosphate.
Lathanum itself is classified as a rare earth element. It's not actually that rare in deposits, it's just fairly difficult to mine and extract, hence why phosphate removers with lanthanum compounds are relatively expensive and largely unnecessary if proper chlorine levels are maintained.
chem geek may have more enlightened details on this but that's the Reader's Digest, probably somewhat correct explanation.