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I just got the news that the New Horizons has passed by some remote planet on the edge of the solar system.

I was surprised that the guy from NASA says that it might take 24 months from us to get the photo of that planet.

The solar system is not that big, right? It is slow because the signal transmission is slow, right? But why is the transmission so slow?

New Horizons has just passed the Kuiper Belt Object (KBO) 2014 MU69 also known as Ultima Thule. KBOs form a belt of asteroids (the Kuiper Belt) from Neptune's orbit outwards and of which Pluto is the largest member of the Belt. During the encounter with Ultima Thule, all of the 7 instruments on New Horizons were gathering data (although not all at the same time) and the total data collected is expected to be about 50 gigabits of data (compared to 55 gigabits of data taken during the Pluto encounter in 2015).

Since New Horizons is about another billion miles further out than Pluto was and 3 more years have elapsed, there is less power for the (tiny) transmitter and the signals are much weaker. The bit rate is about 1000 bits per second and so the 50 gigabits will indeed take about 19-20 months to transmit everything back. The first image at about 300 meters per pixel resolution and so about 100 pixels across the 30 km KBO, should be received on Jan 1. A second higher resolution image with about 300 pixels across the KBO is expected to be downloaded by Jan 2. There will be a press conference on Jan 2 when these images are due to be released and shown. (more details on what to expect when at Emily Lakdawalla's Planetary Society blog entry)

After the initial data download, they expect to perform some analysis to see which images have the best data with 2014 MU69 in the frame. Given the uncertainty in the position of 2014 MU69 and the high speed of the encounter, they had to shoot strips of images and not all will contain the target. These data will be prioritized in the downlink so they arrive on the ground first and can be analyzed first.

On top of the slow data transmission rate (explained in astrosnapper's answer), I think it is worth pointing out that New Horizons will enter solar conjunction next week, meaning that we won't be able to receive any transmissions from it due to the Sun blocking them.

I don't know how many times this will happen over those 24 months, but it is an additional reason for the long(er) wait.

Source: NASA News Conference [42:18]

The other answer mentions it but this gives a bit more theory as to the why.

It's effectively for the same reason that your phone or Wi-Fi don't work as well and slow down when that they are far from the hotspot or cannot get a clear line of access to the cell tower, more commonly known as having "few bars": the signal gets weaker and as a result the signal-to-noise ratio (SNR) goes up. This means that the error rate - failure to successfully transmit a bit and have it received correctly at the sender - goes up, because there is a greater probability that some fluctuation, like other sources of radio waves such as the stars and astrophysical phenomena, or even thermal fluctuation within the receiving devices themselves, can be taken as representing data. As a result, to ensure that the bits successfully make it through, they have to be transmitted for a longer time so that they can be more clearly distinguished over that noisy background and won't be spuriously flipped. The poorer the SNR, the longer you need to transmit to make it clear. Another way to say it is that when you have a noisy background, and you turn on the transmitter, it creates a statistical bias in the noise fluctuations as its transmissions become superimposed upon them, e.g. putting a sinusoidal variation on top. At very low levels, this statistical bias is very small and thus requires a long sampling time to collect enough data to tease it out with high probability and since you don't know what data is coming at you by definition, you want the thing you're trying to tease to be as predictable as possible over the teasing time, thus you must be sending only a single specific type of signal over that time and not switching between bits, limiting the bit rate to exactly that time. A mathematical theorem called the Shannon-Hartley Theorem analyses this precisely and gives the exact bounds on just how fast you can transmit data and still have it reliably heard over a given level of noise relative to the strength of the transmitting signal.

For an understanding of the spatial scales involved here and thus exact what one is up against: your phone has to deal with a cell tower maybe 10 km away ... but here the probes are easily over 6000 Gm away (that's 6000 billion meters and so 600 million times further), and naturally we need a very large antenna, and because of the concerns just mentioned, the transmission rate is limited to, as said, about 1 kbit/s, taking a full millisecond for every bit transmitted, versus your phone at several Mbit/s or more. To downlink an uncompressed 8-bit (greyscale) 640x480 picture at that rate of 1 kbit/s takes 640*480*8/1000 ~ 2500 s or 2.5 ks (kiloseconds). A 4K UHD image would take 3840*2160*8/1000 ~ 66 ks to downlink, or the better part of a day (86.4 ks). Compare that to your broadband domestic Internet where streaming 4K video (up to 60 frames per second so four million times faster) comes down with ease.

This is also one of the reasons that Martian exploration would be significantly aided by, and it has been proposed to use, telepresence robotics controlled from a human base near, but in orbit of, the planet.

ADD: More accurately, the distance to 2014 MU69 is around 6600 Gm.