SpaceX as a company is trying to solve the "transportation problem" of becoming a spacefaring civilization. Specifically, how do you reduce the cost of access by multiple orders of magnitude?
There are three big targets they need to hit:
1. Make Low Earth Orbit (LEO) cheap with a combination of launch cadence and reusability.
2. Reset the rocket equation in LEO by doing in-orbit fuel transfer.
3. Reset the rocket equation at Mars by generating fuel from the environment.
* The reasoning...
(1) A high launch cadence means you can switch from artisanal to mass production. You drastically reduce fixed costs of ground equipment and personnel. Experience translates to increased knowledge, reliability, and safety. You are less prone to schedule slip costs because there are more trains leaving the station.
(1) Reuse lets you get more life out of expensive components. It also helps you hit a higher launch cadence with less manufacturing capacity, further reducing costs.
(2) In orbit propellant transfer lets you start with a full tank again in LEO. It might take four tanker launches to refill one tank, but ultimately you can increase payload to non-LEO destinations by an order of magnitude. Being less mass constrained also makes things simpler and cheaper for payload development.
(3) Propellant production, aka in-situ resource utilization, lets you fully refuel at Mars or the moon. This lets you send non-trivial payloads in the return direction.
* The progress so far...
SpaceX has regular reuse of the 1st stage of their Falcon 9 and Falcon Heavy rockets. They have begun catching the fairings (nosecone) in giant nets mounted on ships and reusing those too. The 2nd stage is not reused. Doing so is difficult because it's much further downrange and has much larger velocity.
SpaceX has used their progress on reusability to drive down costs and capture a large part of the commercial launch market.
SpaceX is launching Starlink satellites to provide a global, low-latency, high-bandwidth internet access. Existing satellite internet is based on geosynchronous orbit satellites which have high-latency and low-bandwidth. SpaceX hoists 60 Starlink satellites per Falcon 9 launch. There are a couple hundred satellites in orbit already with several thousand planned. Their low orbit means they have a lifetime of just a few years and require regular replacement. If Starlink is a commercial success, then SpaceX will have found the "demand" needed to drive a high launch cadence.
* Why the SN-5 hop matters...
SN-5 is a prototype second stage for the "Starship" rocket. This rocket has both a massive payload capacity and _full_ reuse. In conjunction with Starlink this would solve (1). This prototype is demonstrating progress on the manufacturing process and viability of the design. There is still tons of work to be done.
SpaceX is already planning to do in-orbit propellant transfer using Starship, which then solves (2).
SpaceX has not begun on (3). However, the Mars 2020 rover mission actually has an experiment on board which does propellant production at demonstration-scale. The chemical reactions (sabatier process) are pretty straight forward, but providing sufficient energy and gathering the resources is a massive engineering challenge.
Ultimately, SpaceX has a coherent, achievable vision for reducing space transport costs by multiple orders of magnitudes and has made demonstratable progress towards that vision.