Why does WOTS+ need a checksum during signing operation?
To understand why we need a "checksum", we can take a look all the way back to 1979 paper "A Certified Digital Signature" by Raphle Merkle. 

From page 14 on the treatment of the Lamport-Diffie OTS: 


  The method as described thus far suffers from the defect that B can alter m by changing bits that are 1's into 0's. ... However, 0's cannot be changed to 1's.


This is because, if one changes 1 to 0, she/he can simply calculate the image of the hash for that bit and get another valid signature. 

The countermeasure as proposed was, quote: 


  by signing a new message m', which is exactly twice as long as m and is computed by concatenating m with the bitwise complement of m.


The similar method was used in WOTS, except instead of concatenating m with its bitwise complement, it's concatenated with the "additive" complement of the sum of its indicies. 

So this is more like a padding scheme (similar in purpose to that used in RSA), but since it's a number, we'll just call it a checksum. 

As to the update in OP, the checksum also plays an important role in the provable security reduction of WOTS.