The softness is not a Lightroom settings change. It is restricted depth of field, created by opening up the lens' aperture to various degrees.
The strength of this effect appears (!) greater in your second example. To see if you can create it with a typical cheap kit lens, I tried to replicate that sort of scene here, using a 28-105mm f/3.5 lens racked out to its maximum focal length. Your average kit lens doesn't have quite as wide a maximum aperture as this one, but we can stop it down a bit to get the same effect.
Here's the test scene at ƒ/22:
In order to get that framing, I had to use this lens' macro feature.¹ Some consumer-level zooms have this capability, but not all do.
The USB memory stick on the right is about an inch closer to the camera than the toy camera body on the left. The heat sink behind them is about 4 inches back from the toy camera body. In all these photos, I focused on the logo of the toy camera to mimic the effect of your second example photo.
By opening the aperture to ƒ/5.6 — a common maximum aperture for the lowest-end consumer zooms at their longest focal length — we get this photo:
This isn't quite the effect we see in your second scene. The background isn't blurry enough, though that could be because the background is farther away from the foreground objects in that scene than in mine. The foreground is a closer match. In both photos, we can start to see the frontmost parts of the near object get a bit blurry.
Another common maximum aperture for consumer zooms is ƒ/4.5, giving us this stronger effect:
If we open this lens' aperture all the way up to ƒ/3.5, we get this:
That's a stronger effect than your scene shows, but without knowing the distances and sizes involved, we can't actually know whether my test scene actually replicates what is going on in the photo you linked us to.²
You might think that the apparently weaker effect of the first photo means it could be done with most any lens, as long as you have control of the aperture. But surprise, a bit of peeking into the EXIF data shows that it was taken with a Canon 5D Mk II camera and a 16-35 ƒ/2.8L II lens, wide open. That's a professional-level camera and lens, the pairing costing over US $5,000 when new. Why do we need such high-end equipment to achieve this effect? Although full-frame cameras give shallower DoF than the more common sorts of camera, this was shot at a pretty wide angle (30mm). The shorter the focal length, the deeper the apparent depth of field. We need a very wide aperture to counteract this.
The wider the maximum aperture on your lens, the more expensive it will be, all else being equal. This is simply due to the fact that the wider the maximum aperture, the bigger the lens elements in front of it must be in order to get enough light down to the aperture. (Otherwise, you get vignetting.) The more glass you have, the more expensive the lens. Without knowing more about your example scenes, I can't say how far down this road you need to go.
That said, every major interchangeable lens system has something like the nifty fifty, which can achieve results like this inexpensively. You may not be able to get the framing you require with that focal length, however.
There are plugins³ that can help you fake low DoF, but it's much easier to do it in-camera. You really only want to use one of these programs when you've already taken the picture and find that the DoF isn't shallow enough.
Lightroom also has a Blur function, which, combined with the Adjustment Brush feature, can do a poor job of faking a 2-level shallow DoF. The plugins will generally do a better job, as they will be able to get multiple levels of DoF, do a proper lens blur instead of a too-perfect Gaussian blur, keep the near objects' edges sharp while still blurring the background right up to their edges, etc.⁴
For our purposes here, we can define macro as the ability to focus much closer than 10 times the focal length. For a 105mm lens, that means much closer than a meter. A typical macro lens will focus down to approximately 4× its focal length, or about 0.4m in our case here.
In the test scene above, I was nearly at my lens' near-focus limit. The lens couldn't have been much farther away from the nearest parts of the scene than half a meter.
I checked the second photo for EXIF/IPTC data that would answer some of these questions, but it's mostly been stripped out.
I did learn that the photo was last touched by a French-language version of ACDSee, though, which is weak confirmation that we're not dealing with Lightroom trickery here. ACDSee and Lightroom are similar sorts of programs, so that there is no real point to managing the same set of photos with both programs.
There isn't some ACDSee magic going on there, either. Aside from a bit of color and tone curve adjustments, I don't see anything in either photo that you can't do purely in-camera.
Topaz Labs' Lens Effects, the Bokeh feature in Alien Skin's Exposure, or onOne's Perfect Effects. There are probably others.
These three programs also come in standalone forms, so you don't need Lightroom in order to use them.
The easiest way to see that someone is using a simple blur effect to fake shallow DoF is to look at the border between the near object and the background. You may see one of two telltales:
Background blur that spills over onto the foreground object's edges
The opposite problem: blur that doesn't quite come right up to sharp foreground object edges, but instead stops a short distance away, leaving a halo of sharp-ish background around the foreground object(s)
One of the things the above-mentioned commercial bokeh plugins do is help you establish a sharp mask between the foreground and background objects, so that the background blur exactly meets the foreground object's edge.