Why is protein cyclisation desirable?
As described in the abstract of Synthetic Cross-linking of Peptides: Molecular Linchpins for Peptide Cyclization by Derda et al., protein cyclization improves:

resistance to proteolytic degradation and conformational stability. The latter property leads to an increase in binding potency and increased bioavailability due to increased permeation through biological membranes.

In simpler terms, a protein that is cyclized is less likely to:

Fold into an unhelpful shape.
Be broken up by an enzyme.

Thus, they are more likely to:

Act as better binders for a given target
Make it across a cell membrane