“Thy bosom is endeared with all hearts”
'Endeared' here means made more precious or enriched in worth, not simply made dear in the modern sentimental sense. The youth's breast is enriched — as a treasury is enriched — by containing within it all the loves the poet once thought were gone. OED records this sense (enrich, make of greater value) as active in the sixteenth century.
“holy and obsequious tear”
'Obsequious' retains its older Latin meaning here: dutiful in performing obsequies, the funeral rites of the dead. A tear that is 'obsequious' is one offered in ritual mourning — the correct, reverent grief owed to the deceased. The word had not yet narrowed to its modern sense of grovelling or fawning.
“interest of the dead, which now appear”
'Interest' carries both its legal and its financial meanings. As a legal term it means the rightful claim or due that the living owe to the dead — the proper respect and mourning their memory is entitled to. As a financial term (current in the 1590s) it means an increment owed on a debt. The poet has paid this interest — tears as tribute — on loves he thought dead; the sonnet now reveals they were never truly gone but were merely 'removed' and hidden in the youth.
historical The sonnet form“trophies of my lovers gone”
'Trophies' means memorial tokens or commemorative objects, not prizes in the modern sporting sense. In the Renaissance a trophy was an emblem or monument displayed in honor of past victories or relationships — objects hung in a place of honor to keep memory alive. 'Lovers' here means those whom the poet has loved, not specifically romantic partners: friends, companions, people dear to him whom he believed lost to death.