Why did Southerners (and sympathetic Northerners) see Lincoln as such a threat?

This was because they saw him as a "Black Republican." 
They believed that he was going to be bent on abolishing slavery.  They did not believe
that he simply wanted to keep slavery from spreading to new territories.  Instead, they
believed that he would try to abolish slavery even in the South where it already
existed.

Another reason for this was the fact that Lincoln
was not even on the ballot in 10 Southern states.  That meant that he owed nothing
whatsoever to the South in terms of his election.  Southerners and their sympathizers
believed that this would make him even less likely to treat them in any sort of a fair
way.

These attitudes can be seen in the following quote
from the link below:

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Many white Southerners believed that Lincoln's
election meant that a final Northern push to abolish slavery throughout the United
States was just around the corner. Countless furious speeches and editorials reflected
the Southern certainty that the "Yankees," as the Northerners were sometimes called,
would soon begin the process of reshaping Southern society without regard for the
feelings of its white citizenry.