What was mysterious is that on my old router i had to manually do the static routing, on this one, it simply worked once I switched subnets.
You didn't have to do anything because no routing was being done. The switch ports on the routers are incapable of doing any sort of routing. The only layer 3 routing that gets done is between the WAN and LAN ports. This is why a router can be completely crashed and still switch packets -- home "routers" are just four port switches with routing software and an additional interface.
Subnets, networks, all of them act as barriers. When you're dealing with a device that's strictly layer 2, it has no way of knowing how to do any layer 3 switching.
The order that data is pushed about in a standard networking device is defined by the OSI model as follows: (1) physical, (2) data link, (3) network, (4) transport, (5) session, (6) presentation, (7) application. Applications are at the highest level and physical is at the lowest. Whenever you initiate communications between two devices, it'll move down the layers on the source computer and up the layers on the second computer. Encapsulation occurs in this order.
The bits that actually make up the packet are generally contained in the application, transport, network, and data link layers, in that order (there are some additional complexities thrown in, such as packet size limits, but for this example let's just assume that everything is contained in one packet). The application data is surrounded by the network layer data, which is surrounded by the data link layer data, which is sent via the physical layer. Let's say you want to get to initiate a file transfer between your computer and the NAS and we'll assume that your computer has an IP address of 192.168.0.200 and a MAC address of 00-01-23-45-67-89 and your NAS has an IP of 192.168.1.11 and a MAC address of 00-FE-DC-BA-98-76.
At this point, we have the data that you want to send. The data would get wrapped in a TCP packet (transport layer, I believe), to provide the necessary port numbers and other information, then an IP packet (network layer), giving the source and destination IP addresses among other things.
From there, our humble little TCP/IP packet is wrapped up in a frame, the units used in layer 2, the data link layer. This is the layer that the switch works in and is the sole reason that you cannot cross IP networks using a normal switch. The frame consists of the source and destination MAC addresses. Layer 2 devices don't care how data gets from point A to point B -- they just look at the MAC address and figure out how to pass the information from device to device.
This is where the hang-up is happening in your case. Since 192.168.1.11 is part of the 192.168.1.0 network and not the 192.168.0.0 network that your computer is in (since your subnet mask indicates that all three octets represent the network), your computer (correctly) assumes that the device isn't in your network. Since your gateway is set to your router's IP -- no doubt something like 192.168.0.1 -- it directs the frame to that device. Now, our frame is at the router. Since the destination MAC address matches that of the router it grabs and starts unwinding the packet. When it checks layer 3 it would see that the destination address is, indeed, outside of its network. Since it only does routing between the WAN port and the internal switch, it would attempt to find the 192.168.1.0 network using the WAN port. Since that network doesn't exist on the Internet it would simply time out, preventing you from accessing the NAS.
If the NAS were on your network (let's say its address is 192.168.0.11 instead) your computer would recognize that it was on your network right away. Instead of sending it to the default gateway it would first discover the MAC address of the NAS using ARP (address resolution protocol -- a means to get a MAC address when only the IP is known) and when the frame is created it would set the destination MAC address to be the NAS. When the switch receives the frame it would look at the destination address, recognize it, and send the packet off on the appropriate port, getting it straight to the NAS without the layer 3 software even touching it.
See? I told you there was nothing complicated or mysterious about it.
I love networking, can't you tell?