VPN's per se do not bypass firewalls, they "tunnel" through them. Let me try and explain this in more detail.
When your computer wants to get content for e.g. a particular web page, it creates an HTTP request. This packet is wrapped into a TCP packet, the name of the website is resolved to an IP address, and the TCP packet is handed over to the IP layer for routing. The IP layer decides, based on the IP address and its routing tables, where to send the packet (the next hop router, which is usually your default gateway). It wraps the TCP packet into an IP datagram, puts the MAC address of the next hop router into it and hands it over to the ethernet interface, which transmits the whole shebang onto the wire.
Firewalls work at the IP layer of this whole machinery (well, usually they do). Your normal SOHO router/gateway/modem device will have a firewall that allows outgoing connections and any return packets for these.
Now what happens when you establish a VPN connection? The VPN client creates a connection to a VPN server somewhere else. The important part is that it then also alters your routing tables, which generally results in the fact that the IP layer now routes all or some of your outgoing traffic into the VPN client instead of directly out the interface. The VPN client then wraps the entire IP datagram into another TCP packet (and at this precise moment the original packet becomes effectively invisible to the IP layer), and this packet is now sent to the VPN server (which unwraps it and then passes it on).
The net effect of this is that of a "tunnel". Firewall and routing rules which would ordinarily apply to a packet are "bypassed" by pushing the packet through the VPN connection. Which also means that if the VPN tunnel handles ALL your outgoing traffic, then any protection mechanisms that are applied at the SOHO router are now ineffective.
I hope this explains the meaning of "bypassing" in this context.