It is possible to just say "un album photo" in French, which is literally "a photo album". Here, they chose to use "un album de photos", which is literally "album of photos". "De" is a preposition that is used to help define the contents of the album, and since there are many photos in the album, we have to use the plural "photos".
Articles are pretty much always used in French. Here, they are talking about photos in general, so we should be using the indefinite plural article "des". However, as "album de des photos" sounds quite strange, the indefinite article is dropped. It is actually rare to see "de" followed by an indefinite article when it is used to help describe something, or identify content. E.g.: une tasse de thé, le jus d'orange, un verre de vin, un roman d'amour, ... etc.
You don't hear the R in linking the two words? Even if you didn't, to me the woman's voice says has an /uh/ sound (or the sound you make when you are thinking aloud) after the L before album. So even if you just hear /luh albahm/ you know it cannot be le album because of the vowel at the beginning of the word album so the apparent schwa sound before album should give you pause. And you know it isn't l'album because you would not hear so much as a schwa after the L. It would just be /LAhL-bum/. The delay in getting to album after pronouncing the L, to me, is very clear and lets me know it's not l'album
There's a skip in the audio that is not natural, so it's not at all clear that it's supposed to be "leur album" vs. "l'album". Unfortunately, there's really nothing to be done about these issues other than try to memorize where they occur. You can report them, but my experience is that they don't get addressed - I assume that re-recording them is a major logistical challenge, so it falls by the wayside.
Yes, there is a difference in pronunciation. But even if someone did not speak clearly, just as the grammar in English usually helps you decipher what is meant, the same applies in French. Du means "of the" and we all know "an album of the photo" makes no sense so clearly du would not work here.
An online dictionary like larousse.fr is a good resource for learning the pronunciation of words. Not only can you play the sound but you can also note its phonetic spelling. But you need to know phonetic spelling in order to grasp the sound correctly.
De is /də/ and du is /dy/. But that makes no sense to you unless you have taken time to listen to a video like this one that helps you know how to work out the sounds from that spelling:
The schwa sound found in de represented by this symbol ə is heard at 0:28 of that video. De rhymes with "the".
The pronunciation of "u" in words like du is a little trickier. You hear it at 0:43, but yoy may need to look for more videos to help you learn the sound.
Youtube has a lot of videos that explain the "mouth gymnastics" you need to do to pronounce words properly. You just need to make a bit of effort to do some research. Google is your friend too.