chore(doc): update the contributing guide with more details (#25321)

This PR explains how to `Verify your fix by deployment` before you submit a pull request.

When contributors complete their hack on any modules under `aws-cdk-lib`, it's unclear to them how to verify and ensure their fix can successfully synthesize and deploy in a real AWS environment. This PR explains how to do that with more details including:

1. How to write a minimal cdk App with your fix to verify it in your AWS account
2. How to run all unit tests against your hack
3. How to run a single unit test against your hack
4. How to run all integ tests against your hack
5. How to run a single integ test against your hack

With the additional content, contributors will be able to iterate their development easily and ensure their hack can successfully deploy in their own AWS accounts as expected before they submit their PRs.

Closes #25196

----

*By submitting this pull request, I confirm that my contribution is made under the terms of the Apache-2.0 license*

CONTRIBUTING.md

@@ -332,6 +332,110 @@ $ cd packages/aws-cdk
 $ yarn watch & # runs in the background
 ```
 
+#### Verify your fix by deployment
+
+If your PR updates a specific library, you might want to write a simple CDK application and make sure it synthesizes and
+deploys correctly. For example, if you modify files under `packages/aws-cdk-lib/aws-eks`, you can write a simple CDK app in typescript to verify its behavior:
+
+
+```console
+$ cd packages/@aws-cdk-testing/framework-integ/test/aws-eks/test
+```
+
+Create a `sample.ts` like this:
+
+```ts
+import {
+  App, Stack,
+  aws_eks as eks,
+  aws_ec2 as ec2,
+} from 'aws-cdk-lib';
+import { getClusterVersionConfig } from './integ-tests-kubernetes-version';
+
+const app = new App();
+const env = { region: process.env.CDK_DEFAULT_REGION, account: process.env.CDK_DEFAULT_ACCOUNT };
+const stack = new Stack(app, 'my-test-stack', { env });
+
+const cluster = new eks.Cluster(stack, 'Cluster', {
+  vpc,
+  ...getClusterVersionConfig(stack),
+  defaultCapacity: 0,
+});
+```
+
+Run `yarn watch` or `npx tsc --watch` in a separate terminal to compile `sample.ts` to `sample.js`:
+
+```console
+$ cd packages/@aws-cdk-testing/framework-integ
+$ yarn watch
+or
+$ npx tsc --watch
+```
+
+Make sure you have configured [AWS CLI with AWS Authentication](https://docs.aws.amazon.com/cdk/v2/guide/getting_started.html#getting_started_auth) as we will deploy it in our AWS account.
+
+Deploy the sample app:
+
+```console
+$ cd packages/@aws-cdk-testing/framework-integ
+$ npx cdk -a test/aws-eks/test/sample.js diff
+$ npx cdk -a test/aws-eks/test/sample.js deploy
+```
+
+This allows you to iterate your development and ensure a minimal sample app would successfully deploy as you expect.
+You have the freedom to interact with it just as a common CDK app such as viewing differences with `npx cdk diff`
+or pass context variables with `npx cdk deploy -c`. You can rapidly iterate your testing with repeated deployments 
+by importing existing resource such as existing VPC. This can save a lot of time and help you focus on the core changes.
+
+```ts
+const vpc = ec2.Vpc.fromLookup(stack, 'Vpc', { isDefault: true });
+```
+
+As this is for testing only, do not commit `sample.ts` and `sample.js` to your PR branch.
+
+Alternatively, you can write this test as a new integration test like `integ.my-test.ts` and deploy it
+using `yarn integ --no-clean`. This may be useful when you need to publish a new 
+integration test:
+
+```console
+$ cd packages/@aws-cdk-testing/framework-integ
+$ yarn integ test/aws-eks/test/integ.my-test.js --no-clean --update-on-failed
+```
+
+After verifying your work with a simple deployment as above, you need to ensure your change can pass all existing
+unit tests and integ tests and fix them if necessary.
+
+Run all the unit tests for a specific module(e.g. aws-eks):
+
+```console
+$ cd packages/aws-cdk-lib
+$ yarn test aws-eks
+```
+
+Or run a specific unit test
+
+```console
+$ cd packages/aws-cdk-lib
+$ npx jest aws-eks/test/name.test.js
+```
+
+Run all integ tests for a specific module(e.g. aws-eks):
+
+```console
+$ cd packages/@aws-cdk-testing/framework-integ
+$ yarn integ --directory test/aws-eks/test
+```
+
+Or run a specific integ test:
+
+```console
+$ yarn integ --directory test/aws-eks/test/integ.name.js
+```
+
+See the [integration test guide](./INTEGRATION_TESTS.md) for a more complete guide on running
+CDK integration tests.
+
+
 ### Step 4: Pull Request
 
 * Create a commit with your changes and push them to a