Measuring application performance in Swift using Sentry's Custom Instrumentation allows developers to track big data-handling functions, measure metrics like throughput and latency, and display error impacts across multiple services. By setting up performance monitoring, developers can configure trace sample rates and implement custom transactions to measure specific function execution times. Child spans enable more granular measurement by attaching mini-transactions to the main transaction, allowing for detailed analysis of individual steps within a function. Additionally, binding transactions to the current scope enables measuring complex functions with multiple branches. With Sentry's Custom Instrumentation, developers can gain insights into their app's performance and improve it by identifying slowest parts of the code.