From what I understand, TunnelBroker is a service that acts as your ISP and allows you a public IPv6 address pointing directly to your machine even if it is behind a NAT.
Well, yes. You can also call it a kind of a VPN, basically. However, the bit about "...even if it is behind a NAT" is somewhat misleading:
These IPv6 tunnels are not meant as a NAT bypass method for individual devices; they're meant to deliver IPv6 connectivity to your router, and from there to the whole LAN at once. (Which indeed works just as if it were provided by your real ISP.)
Once the entire LAN has IPv6 connectivity, there's no "...even if" in that statement anymore. The devices simply aren't behind a NAT in the IPv6 network.
(NAT is specific to each network protocol, so even though your LAN devices are still behind a NAT in the IPv4 network, they're simultaneously not behind a NAT in IPv6.)
TunnelBroker claims to give you a free /48 block of IPv6 addresses. That's >65k addresses
That's >65k subnets, each having a practically-infinite amount of addresses, due to the full IPv6 address length being 128 bits and the "standard" subnet size being /64.
How do I use them?
You use a router for that. (As mentioned above, either configure the tunnel on a router, or turn the tunnel endpoint into a router if it's capable of it.)
The tunnel configuration is the same as before: the device still has its own address (the "Client IPv6 address" from tunnel info), it just additionally receives traffic meant for the extra "routed /48" network.
You don't need to change the tunnel on your side for this to happen; it's enabled by the ISP (Hurricane) updating the routing table on their endpoint.
(You can even completely ignore the fact that it's a tunnel; everything works the same way as if you were receiving the IP connectivity over a plain Ethernet cable. The only difference from, say, native ISP-provided IPv6 is that you don't need to use DHCPv6-PD to begin receiving the traffic.)
Once the router begins receiving traffic meant for the new network, it gets to decide where to send it further. Usually you start by subdividing the /48 into standard /64-sized subnets, and just assigning one of them to the LAN interface or to some VLAN. For example, if you had
2001:db8:123::/48, start with assigning
2001:db8:123:1::1/64 on your LAN interface.
(This part is nearly identical to configuring networks in IPv4.)
Finally, set up an address auto-configuration service for your LAN. Instead of DHCP, in IPv6 the primary autoconfiguration mechanism is SLAAC, i.e. "Router Advertisements".
Besides TunnelBroker (also known as HE.net from the company), chances are that you'll find many tutorials about configuring a router for "6to4" or other services. They all work the same as far as LAN usage is concerned. Here are configuration examples for Mikrotik RouterOS, for Windows XP, for Gentoo Linux, for generic Linux.