כל דכפין ייתי וייכול, כל דצריך ייתי ויפסח
Apparently, this is the same concept as inviting guests, but really expressing two ideas. One idea is inviting the hungry to come and eat. The second idea is inviting those that are not hungry but lack the needs of Pesach (or of Yom Tov) such as wine or meat. Thus, the word ויפסח, is a word derived from the word to make the Pesach holiday (and not the Pesach sacrifice since applies even outside of the Land).
The reason for the proclamation at the night of the Seder (and not earlier) can be understood according to the Gemara (Taanis 20b). The custom of Rav Huna was to purchase all the vegetables from the market on Erev Shabbos at the end of the market day, and then, he threw them into the river, not giving them to the poor. The stated reason was for the poor not to rely on handouts and therefore, would not make proper preparations for Shabbos. Thus, one could also expect the same for Pesach preparations (just like Shabbos), and thus, at time of Seder, we need to invite those poor that did not have the means to prepare for the extra needs of Pesach.
This is the place also to explain why we do not make this announcement prior to making Kiddush. It is possible to explain according to the Gemara (Pesachim 106a) on the verse “Remember the Shabbos day”, that remembrance means to make Kiddush as soon as possible at the start of Shabbos. Similarly, about Pesach, the verse (Parshas Bo 13,2) states “Remember this day that you went out from Egypt.” Thus, the remembrance needs to be with Kiddush as soon as possible with the beginning of Pesach.
Therefore, one should not delay to make Kiddush while awaiting the arrival of invited guests (such as the poor). The guests do not have a set time and besides they can make Kiddush themselves when they arrive. Thus, we first make Kiddush and then invite those that do not yet have a place.
