The explanation in the show, given, is that when SG-1 steps through the gate they are in both time periods at once causing the gate and gate room to be there, but none of the people and then it disappearing, revealing the 1969 period location.
As mentioned other places, later, in "2010" it is shown a message is sent back through time from the 2010 location to the 2000 location of the gate, contradicting the events of "1969" where the time is changed, but location is not. From an out-of-Universe point of view this is likely just because they needed to do that for the stories. From an in-universe POV there are several distinct differences between the 2010 gate, the 2000 gate, and 1969 gate (also the other time period shown in "1969" which we just randomly assume is the SGC as well) and that is in 1969 the gate is essentially buried, both of them, and as such there is no "gate" in those time periods. In 2000 they are not using a DHD and they are likely using a significantly less powerful power source than in the 2010 gate which is using a DHD.
What we are lead to believe is that the gate sends out the wormhole and it's bent back on itself, causing the time traveling, and targets the nearest stargate, which is itself in a different time. However, this makes no sense, so I propose that what is really going on is that the wormhole is going around the gravity well and then targeting the gate, but this traveling around is not the cause. The cause is targetting the same gate which since it is already open moves to the nearest gate or place it can open to given the power its been sent there with. The event we see in "1969" is a matter there but there caused by there being enough power to send them temporally 30 years in the past, the gate having already connected, but there being essentially no gate at that time to connect to which causes a "Schrodinger's cat" effect. This then explains why the 2010 to 2000 travel even doesn't produce this effect, and it is simply that there is a gate to connect to on the other side.
I would also guess that the 2010 gate could not open a 1969 gate due to the DHD which would read no connection.
Part of this explanation comes from the understanding that the stargate is moving through time and space all the time and connects to the gate that is closest to its position exit point coordinate. Usually that is a spacial change, but due to targeting the same place the space distance transforms into a time distance and where no gate is present, because the lock on already happened, the time-space is duplicated briefly which is somewhat explained in the Tollan episode. Daniel is shown that how they communicate is they bring 2 points together and he is shown this with a stick being bent. When he talks about bending space-time (which is how the stargate wormhole supposedly works) he is told with a sigh, "no". Assuming the Ancients are superior to the Tollans, they'd understand whatever principle was being stated and would likely use the same, or similar, to produce the effect with the stargate. This, along with the episode itself talking about the wormhole being bent, and another episode ("Red Sky") showing that matter can exit the wormhole, before getting to the stargate, and even how the stargate works in general shows that it's not actually a wormhole.
So TL;DR version: The gate is already accessing time and space. When the gate is locked onto another gate that's open, or there is too much energy, it shunts it to the closest spacially close gate in the "same" location. If there is no other gate spacially, then it converts the energy and shunts it temporally causing time travel. The gate is always creating the effect that we see in 1969, but because there is another gate there we don't experience it, and this is why they're able to reintegrate without a second gate. Because the same gate is acting as both outgoing and incoming at that moment.