Just as Shabbat is a day of rest and delight after a week of hard work, so is Olam Haba a time of rest and delight after a lifetime of hard work.
After we work all week and prepare for Shabbat, on Shabbat itself we rest, enjoy the world Hashem created (without interfering with it), delight from Shabbat-specific activities (nicer meals, great company, increased Torah study) and benefit from all the work done during the week.
Similarly the Olam Haba is a world of rest, delight and enjoyment from a life of work. During all our life we work to perform mitzvot, sanctify the name of Hashem, change the world for the better and improve our neshama (soul). In Olam Haba our neshama will rest (since it cannot perform any work) and reap the rewards from all this work by enjoying closeness to Hashem. This is the meaning of the quote from Pirkei Avot in your question (which incidentally is not directly related to Shabbat but at our entire life).
A wonderful and highly readable book expanding on these ideas is Stop Surviving, Start Living by R Ben Tzion Shafier
PS. I couldn't find your other quote
This world is like the eve of Shabbat, and the Olam Ha-Ba is like Shabbat - I think it is paraphrased from Avoda Zara 3a
Whoever toiled on the eve of Shabbat will eat on Shabbat but whoever did not toil on the eve of Shabbat -- from where will he eat on Shabbat?
which Rashi interprets to mean that whoever performed mitzvot in this world will be rewarded in the olam haba