Seven is significant because during the onset of creation, seven was the number of surrounding sheaths in the Yajna (sacrifice) in which Purusha himself was the offering. And, the number of such offerings (Samit) was also 21, a multiple of 7.
The relevant mantra from the Purusha Suktam is the following:
SaptAsyAsan paridhayas (1)
Trih sapta samidhah kritAh (2)
yat yajnam tanvanA (3)
Abadhnan purusham pashum (4) ||
The Gods, giving body to yajna (3), bound Purusha, the seer (4). Seven were
the surrounding sheaths (paridhi) (1), Thrice seven the prepared fuels
Rig Veda 10.90.15
As per Rishi Aurobindo, the seven paridhis being talked about here are the seven planes of existence.
Three lower ones
Matter (anna), life-energy (prana) and mind.
Three higher ones:
Ananda (jana), conscious-force (tapa) and truth of existence (satya).
The seventh plane is the link-world between the two triplets and is
called Mahas, the plane of Vijyana or Super-mind. The twenty-one fuels
are the three energies in each of the seven planes.
So, as we can see the number 7 and its multiple (21) had significant roles to play during the onset of creation itself. So, that's basically why it's important.