The most typical Japanese comedy style (漫才) consists of
- Someone saying/doing a silly thing (ぼけ) and
- Someone pointing that out (つっこみ).
(Confer my answer to this question.)
ぼけ is the joking part as seen in jokes in many countries, but one difference form the commedy style often seen these days in Western countries is that the ぼけ person himself/herself does not make it explicit that it is a joke. Rather, they pretend that they are not aware of the funniness of what they are saying or doing (In Western commedy, a counterpart will be something like Mr. Beans). The ぼけ person makes jokes at certain points in the talk, where the つっこみ person is expected/supposed to pause the talk and point out or explain to the audience what the joke was. These points are
突っ込みどころ "points where the 突っ込み is supposed to be done".
満載 means "loaded full of". It is used here metaphorically. Following Yadokari's suggestion, the English counterpart of the person with the つっこみ-role will be "the straight man".
満載 is the predicate predicated of its argument
突っ込みどころ, which is a compound noun consisting of
突っ込み （verb stem used as a nominal） and
ところ "place" (which underwent sequential voicing (連濁) to become
Recently, this expression has evolved into meaning, outside of the commedy context, that someone's talk or act is so silly or stupid that there are full of points where one can point that out.