"The seventh year shall be a Shabbat Shabbaton for the land, a Shabbat to the Lord, your field you shall not sow and your vineyard you shall not prepare." (Vayikra 25:4)
The Sifra on this verse comments that the reason the word Shabbat was applied to the Shemittah year was to compare it to the weekly Shabbat because both are referred to with the phrase "to the Lord".
Tha Malbim, commenting on this Sifra, note that while the word Shabbaton is applied by the Torah to all the holidays, the word Shabbat itself only appears in reference to Shabbos itself and Yom Kippur. By the same token, the phrase "a Shabbat to the Lord" only appears in reference to Shabbos and the Shemittah year since both remind us that God rested on the seventh day and testify that there is renewal in the world. It is fitting, says the Malbim, that just as animals and people get to rest on one type of Shabbos that there should be a Shabbos for the land to rest too. Why does the land need a special year for this? By all rights, just as everything else rests on Shabbos, so too should plants and other products of the land stopped growing every seventh day but since that would violate the laws of nature which God created (and which He does not wish to violate) therefore the seventh years was chosen to make up for the six years where the land could not observe Shabbos. This is why in parshas Mishpatim both the weekly and the yearly Shabbos are brought next to each other - the latter is an extension of the former.
Further, the Malbim reminds us that in some places in the Torah the year is called yamim, days, since there are two ways to look at the sun's movements in the sky. The first is the 24 hour cycle where the Earth revolves around and the other is the way the sun moves back and forth in its position in the sky over 365 days until it returns to the same spot it started, another form of "day". There are therefore the two terms: yimei haadam which refers to the 24 hour day that our lives revolve around and yimei ha'aretz which refers the the complete cycle of the Earth revolving around the sun. The yovel then brings a 50 year cycle which completes the "days" of the land which is why everything "resets" at that time in terms of land ownership.
The final example of this type of cycling that the Malbim brings is the gemara which tells us that the Earth will endure for 6000 years and lay waste for another 1000, paralleling once again the 6 days of Creation and the seventh day being Shabbos. However, here he brings what I find is a very clever bit of math.
In 6000 years there are a potential of 120 yovel years. By amazing coincidence, we are told at the beginning of parshas Noach that the ideal potential life span of man is 120 years. Therefore, the Malbim concludes, we see the ultimate connection between God, for whom a day is 1000 years as brought in Tehillim, and man. Both have 6 days of creativity and work and the seventh day for each is one of rest. This is how the Shemittah therefore ties into Shabbos Bereshis.