Translation:He put out all his shirts on the bed.
It should be acceptable, though I don't know if it has been added yet. 'Put out' in English is a separable phrasal verb, which means the word 'out' can optionally be separated from 'put' and placed after a noun direct object, and in fact, must be placed after a pronoun direct object. Thus, 'He put out his shirts ...,' 'He put his shirts out ...,' and 'He put them out' are all grammatically possible, but not 'He put out them.' There are a number of similar expressions that one sees in DL, such as, 'turn on/off the TV (OR HERE on/off).' Duo just needs occasional help from us reporting legitimate alternatives. Part of the problem is that, many phrasal verbs are not separable. You can't 'run somebody into at the mall'.
Notice regarding the TTS: the overwhelming majority of Swedes don't say "lade", but rather just the abbreviated "la". Even though the latter is mostly not regarded as correct in formal writing, it's gaining ground, and it's only a matter of time until they'll be perfectly acceptable both. (Feb 18th, 2015)
It really just means he put them on the bed. The "fram" could imply some displaying - he may have arranged the shirts one by one, rather than stacking them; but it's more likely that they were simply in some wardrobe or similar and the "fram" means he took them out in plain sight.
'lägga fram' has the meaning of 'present' and 'lägga ner' has the meaning of 'lay down'. "Han lade fram alla sina skjortor på sängen." - he put the shirts out for presentation to himself (possibly). Another example is "Man får lägga fram kläderna så de kan hitta de ." = "You have to present the clothes so they can find them." However, I have seen that these phrasal verbs have sometimes an abstract meaning like: "Staden har inga planer på att lägga ner bion." = "The city has no plans to close down the cinema." "Vi har försökt lägga fram ett förslag.." = "We have tried to put forward a proposal .." I am not a native speaker so any corrections are welcomed.