So (and this is an Edit after reading some of the great and informative answers and comments) - A receiver circuit should be simple to build and connect the analyzer into it? What about software to 'understand' the IR codes?
My preference is to burn an mcu for this rather than try to use interrupts but either way you need to find the pattern and decode it and there will be noise in the room other remote commands, etc.
Once you get this powered and are examining the signal with a scope or your logic analyzer, is your receiver inverted or not. The signal is AM radio like but square. Ones lets say are a burst of the carrier frequency, receivers like this will just give you a pulse roughly the duration of that burst, but if inverted that might be at gnd level not vcc. so figure that out. Then look at the NEC protocol for example a number of them are similar with different sized sync patterns. For NEC this says that the sync pattern is 9ms followed by 4.5ms, then it goes into 32 bit patterns. The signal level that is the same as the 9ms (lets assume high or vcc) is a fixed length, the space between is what varies and that tells you a zero from a one. this pattern ends with a short pulse, very clean pattern. If you have a programmable remote you can cycle through till you find one that matches or is similar. Sony SIRC is similar, has a sync pattern you can search for but the gaps are fixed and the pulses vary. Also very easy to decode for a first time project.
The software or logic if you use a cpld/fpga, would in these cases look for that long sync pulse, if you made a program to only do that, and blink an led or output something on the uart whatever to indicate you found that pulse it should line up with pressing the button on the remote. understand as you hold the button it repeats (and the repeat pattern is sometimes not just the same pattern over and over again some schemes have a separate repeat pattern from the initial code). then you look for the gap after the sync pattern and so on.
If you look at philips recs80 there might not be a simple sync pattern to search for out of the noise, but instead you have to go right into decoding and hope you received the whole thing. Some patterns have a way to confirm you got the pattern (the second half is the code inverted for example) some dont other than it matches one of the values you were looking for. Worse look at philips rc5, that is good and bad that encoding is biphase-L sometimes just called biphase sometimes called manchester, you are insured that the data is either a half bit cell or a whole bit cell. so you are looking for pulses and gaps of those two lengths, and then you have to sort out the code. If rc5 always starts with a one, then you are either looking for half bit cell or whole bit cell length pulse depending on whether the code starts with 11... or 10... with pencil and paper you can figure out if this gap/pulse was long and the prior was blah then this bit was X (one or zero). or perhaps better save the data as half bit cells starting with a 01 for the leading one. then look at ever other the second half bit cell is the data value 011001... x1x0x1 is 101.
someone or several dozen folks in the Arduino world will have code out there to deal with various protocols and you can just use it. if that were your goal you wouldnt have needed to buy any equipment. It makes a difference what frequency your led uses (as well as the receiver) as to how well it is going to work but clearly universal remotes dont have more than a couple leds and I doubt they are different frequencies. You will need to generate the right carrier frequency which is shown in the link above, I hate putting links in answers it is very bad form but you can google/find these on your own. If you get it wrong then it will still work you just cant walk around the whole room and get the TV to work you have to be within a certain distance plus or minus. a carrier frequency and an and gate you can in theory take the signal you are seeing from your receiver and drive this carrier and gate thing and feed that to the led. With a cpld or some mcus you can do the work in that device and drive the led directly. Might want to drive it through a transistor to get more power, but do your system engineering and the math that would go with it based on the specs for the leds that are available. Some mcus I know of a family of them from ST (some of the STM32s) that have an IR timer, which basically has the and gate in the chip, you use one timer to generate the carrier and another the data pattern. I have been able at times to "count out" so many timer ticks and change the gpio/led state to generate the carrier, then dont do that for a while to generate the gap, repeat. very hard to get it right with many mcus having non-predictable execution times of instructions but something like an old pic16, hand tuned code you can get it exact even with the if-then-elses needed to deliver the code.