I was externally hired by a company to help them with software development. All people at the client are software developers too. I was asked to lead them, but I don't know how. I'm quite introverted, don't talk that much and have a hard time at the client to understand their business processes.
What can I do to learn/develop my leadership skills in the workplace?
I feel sorry for your situation but it sounds like they sent the wrong person to the wrong place at the wrong time.
Speak to your superior to clear things up, maybe they have a different client/gig for you that suits your strengths and capabilities.
In regards to: "What can I do..":
Take management and leadership courses, get yourself a scrum certificate i.E., maybe train some interns or get a small project to get used to leading/managing.
"I'm quite introverted.." Reconsider if leading a team or managing a project is something you really want to do and you if are comfortable with it. There are lot of positions/roles in IT where other skills are required, and since you have 10 years of XP on your record you could find something different without too much hassle.
For a leadership-role you would need to find ways and learn how to become more extroverted, outspoken to voice your opinion, set disciplinary meassures, pat people on the back, speak to clients, make presentations and so on..
I'm quite introverted, don't talk that much and have a hard time at the client to understand their business processes.
What can I do to learn/develop my leadership skills in the workplace?
Based on the first sentence, it could be quite a challenge in the short term. Your best bet is to talk to your boss and see if they can help, perhaps by being a mentor.
In the longer term, consider taking some business, management, or project management classes at a local university. That can at least expose you to some of the ideas and skills necessary to be a leader and perhaps develop your abilities.
You haven't provided much information about the nature of the job - the size of the team, whether you're working remotely or in person, whether you're helping start a fresh project or joining late to help finish the last leg of a longterm project. You also haven't mentioned any specific leadership expectations that the client has given you. This makes it more difficult to answer your question.
In the contracting world, "lead" can mean different things to different people. Expectations might vary widely depending on the team, whether you're working remotely, and (if working remotely) how the team communicates.
I mostly work remote. I've worked jobs where there's a daily stand-up meeting via video conferencing and taking the lead involves some amount of leading these meetings and managing each team member's tasks - not a good fit for an introvert. At the other end of the spectrum, I've worked jobs where the team communicates intermittently over email or Slack, and "leadership" mostly involves writing up periodic code reviews and occasionally answering another team member's question - nothing requiring a very social personality. I've even worked on one unfortunate gig where the client brought me in late in the game to lead the existing team to meet a delivery date, but the other team members (who'd been on the project since the beginning) were so offended by this that they mostly refused to speak with me, so my "leadership" amounted to doing most of the work myself and occasionally merging pull requests.
What can I do to learn/develop my leadership skills in the workplace?
Start with things you know well. Look for places where you can offer advice based on your own knowledge and skills - for example, "hey, I see that you're doing it this way, but I have found that it's faster/easier/better if you do it this other way instead". "In this past I've used this tool/library". Once you're comfortable with giving advice, that can evolve naturally into giving directions.
Look for opportunities to cooperate with others by splitting tasks in a way that plays to everyone's strengths. For example, "hey, I'm working on the backend, but you have more database experience than me. Could you work on the database while I work on the API?" Again, this can evolve naturally into giving directions.
Never try bluff or bluster your way through something just because that's what you think a leader should do. If Sarah's good with MySQL and you don't know the first thing about relational databases, it's fine to ask her "how would you build this?" or say "can you please build this database", but don't try to tell her how to build the database if you don't know what you're talking about.
