What is a spiral arm?
The reason that the Sun, in principle (but see below), may cross spiral arms is that galactic spiral arms are not rigid entities consisting of some particular stars; rather they are "waves" with a temporary increase in density. An often-used analogy is the pile-up of cars behind a slow-moving truck: At all times, all cars are moving forward, but for a while, a car behind the truck will be moving slow, until it overtakes and speeds up.
Similarly, stars may overtake, or be overtaken by, the spiral arms. Inside a certain distance from the center of the galaxy called the corotation radius ($R_\mathrm{c}$) stars move faster than the arms, while outside they move slower.
Since the stars and the interstellar gas follow the rotation of the arms for a while once they're inside, the density of the arms is higher than outside, but only by a factor of a few (e.g.
Rix & Rieke 1993).
When interstellar gas falls into the potential well it is compressed, triggering star formation. Since the most luminous stars burn their fuel fast, they will mostly have died once they leave an arm. Hence, what we see as spiral arms is not so much the extra stars, but mostly due to the light from the youngest stars which are still inside the arms. Since most luminous also means hottest, their light peaks in the bluish region — hence spiral arms appear blue.
Origin of the spiral structure
At least the most prominent spiral arms (especially grand designs) are thought to be created by these long-lived, quasi-stationary density waves (Lin & Shu 1964). The reason that the density waves exist in the first place is not well-understood, I think, but may have to do with anisotropic gravitational potentials and/or tidal forces from nearby galaxies (e.g. Semczuk et al. 2017).
But in fact even small perturbations may spawn gravitational instabilities that propagate as density waves. In computer simulations of galaxy formation, even numerical instabilities may cause this, so the fact that your simulated galaxy has spiral arms doesn't necessarily mean that you got your physics right.
When the luminous and hence massive stars die, they explode as supernovae. The feedback from this process, as well as that exerted by the radiation pressure before they die, may help maintaining the density waves, at least in flocculent galaxies (Mueller & Arnett 1972). Perhaps this so-called Stochastic Self-Propagating Star Formation may also initiate the density waves (see discussion in Aschwanden et al. 2018).
The rotation speed of the material in the galactic disk is roughly constant with distance from the center (this is mainly due to the dark matter halo hosting a galaxy). Hence, stars close to the center complete a revolution faster than those farther away. In contrast, the spiral pattern rotate more like a rigid disk such that, in an intertial frame, the pattern can be described by a constant angular speed $\Omega_\mathrm{p}$ throughout the disk. However, note that spiral arms are transient phenomena; they appear and disappear with lifetimes of the order of (a few) Myr (e.g. Grand et al. 2012; 2014). Sometimes you also see multiple spiral patterns propagating with different velocities.
The Sun in the Milky Way
Note: This section first contained errors based on dubious values for the angular speed of the spiral arms, as pointed out by @PeterErwin and @eagle275.
In the case of our Sun, we happen to be located very near the corotational radius $R_\mathrm{c}$; we sit at a distance of $R_0 = 8.32\,\mathrm{kpc}$ from the center of the galaxy (Gillessen et al. 2017), while
$R_\mathrm{c} = 8.51\,\mathrm{kpc}$ (Dias et al. 2019).
Using Gaia data, Dias et al. (2019) find a pattern angular speed $\Omega_\mathrm{p} = 28.2 \pm 2.1\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{kpc}^{-1}$. At the location of the Sun ($R_0$) this implies a pattern speed of $\simeq235\,\mathrm{km}\,\mathrm{s}^{-1}$. This is only a little bit slower than (and in fact statistically consistent with) the Sun's velocity of $239\pm5\,\mathrm{km}\,\mathrm{s}^{-1}$ (Planck Collaboration et al. 2018). If the spiral arms were "permanent"*, the timescale for the Sun crossing an arm would hence be gigayears.
However, because as described above they're quite transient, we might once in a while overtake a spiral arm. I haven't been able to find firm evidence for whether we have or haven't crossed any arms; Gies & Helsel (2005) argue that we have crossed an arm four times within the last 500 Myr, but base this on matching glaciation epochs with passages through spiral arms (and admit that this requires a lower but still acceptable pattern speed).
The article you link to
I now wrote to Jesse Christiansen (who the linked article quotes) and asked her if she knows whether or not we are moving in and out of spiral arms; she replied within roughly 8 seconds, tagging Karen Masters who chimed in even faster — they both agree that this is an ongoing debate with no conclusive evidence.
Anyway, the article seems to have misunderstood the tweet from Jesse Christiansen. In her animation she shows the journey of the Sun, but shows the galaxy itself as being static, which she did on purpose to keep it simple. Hence, you see the Sun traversing the arms unnaturally fast.
