Packets travel through internetworks, and take many routes through internet routers. On each route that forwards traffic to another until reaching the ultimate destination, what stops them from viewing the packets they receive/forward?
Short answer : you can't prevent them from sniffing your traffic, but you can make it meaningless for them by using encryption.
Either use encrypted protocols (HTTPS, SSH, SMTP/TLS, POP/TLS, etc.) or use encrypted tunnels to encapsulate your unencrypted protocols.
For example, if you use HTTPS instead of HTTP, the content of the webpages you fetch will not be readable from those routers.
But remember that they can still save the encrypted packets and try to decrypt them. Decryption is never about "can or can't", it's about "How much time does it take". So use ciphers and key lengths suitable for the degree of privacy you need, and the "expiration time" of the data you want to "hide". (meaning if you don't care if someone gets it a week after the transmission, use a strong protocol. If it's an hour, you can lower the key length)
In principle, "packet sniffing" is not a job of routers. Routers don't inspect the content of a packet, they just pick the header of the packet and find a suitable exit interface so that it will reach the destination stated in the packet's header.
However, your packets could be sniffed by anyone interested while the packets are moving. Interested bodies (often called attackers) can stop your packet and sniff it before it reaches the destination. This is called active sniffing, and is performed by firewalls. Another way of sniffing is passive sniffing. This method involves collecting copies of your packets passively. Wireless sniffing is a popular attack method in this category.
You can reduce such risks by using encryption techniques discussed above.
The real answer: Nothing you could do to prevent a router to sniff all of the traffic that flows through it. But that should not be the question - this should be: "What could I do to prevent anybody to read my traffic?" - The answer: "Use only encrypted communication/protocols!". The best thing: Use a VPN between you and your communication partner. For all others: HTTPS instead of HTTP, IMAPS instead of IMAP and so on.
As it was previously mentioned, it is not possible to stop a router from sniffing your data.
There is a simple way to make the listener's life harder by using Tor bundle.
Tor is a network of virtual tunnels that allows people and groups to improve their privacy and security on the Internet.
It cannot guarantee that your information will be 100% secure but if you use it properly (e.g Don't enable or install browser plugins), your data should be a bit more safe.