They have no issues with connectivity, they might have higher latency and lower bandwidth than some of the higher end providers. But since +90% of the data transfer on most sites is static assets, using a CDN to distribute those will reduce the bandwidth requirements on the server.
Of course, if you're working on a larger project with enough budget, services like Akamai provide methods of routing your static content via their network, further reducing the problem.
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How do CDN's solve bad connectivity of your app server?
They have no issues with connectivity, they might have higher latency and lower bandwidth than some of the higher end providers. But since +90% of the data transfer on most sites is static assets, using a CDN to distribute those will reduce the bandwidth requirements on the server.
Of course, if you're working on a larger project with enough budget, services like Akamai provide methods of routing your static content via their network, further reducing the problem.