As nobody better qualified than I am has responded yet, I can offer a partial answer (but I’m by no means an expert).
This is from the lay version of the Nimenrix EPAR:
Nimenrix contains small amounts of capsular polysaccharides (sugars from the outer coat) extracted from the four groups of the N. meningitidis bacterium: A, C, W135 and Y. These have been purified and then ‘conjugated’ (attached) to a protein carrier called tetanus toxoid (a weakened toxin of tetanus which does not cause disease, also used in tetanus vaccine), because this improves the immune response to the vaccine.
So, as far as I understand that:
The vaccines that didn’t work were purified polysaccharides. The were replaced with vaccines that joined the polysaccharide to a random protein: when joined that way, the subject develops immunity to the polysaccharide. Right?
It sounds like the thing that it’s linked to is acting like an adjuvant, recruiting immune system components to the injection site where they can familiarize themselves with the polysaccharide and tool up against it. Vaccines with adjuvants are so formulated to make them more effective than the version without. (Aside: it was speculated that the attenuated virus that was used in the AZ COVID vaccine was also acting like an adjuvant.)
A polysaccharide is a starch?. Presumably, the bacterium produces and feeds on this starch?
Nope, it’s part of the cell wall.
Why would adults develop an immune response to a starch? Why did children not develop an immune response to a starch?
Not a clue.
How does developing an immunity to the polysaccharides provide immunity to meningococcus and pneumococcus?
By training immune components to identify and target the cell wall.
You can link to the full EPAR from the lay version, but it doesn’t provide much more useful information so far as I can see.
Here it is.
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ETA - I chose this EPAR entirely at random. Others might be more helpful.